Delay Equations: Functional-, Complex-, and Nonlinear Analysis - Tapa blanda

Sinopsis

0 Introduction and preview.- 0.1 An example of a retarded functional differential equation.- 0.2 Solution operators.- 0.3 Synopsis.- 0.4 A few remarks on history.- I Linear autonomous RFDE.- I.1 Prelude: a motivated introduction to functions of bounded variation.- I.2 Linear autonomous RFDE and renewal equations.- I.3 Solving renewal equations by Laplace transformation.- I.4 Estimates for det ?(z) and related quantities.- I.5 Asymptotic behaviour for t ? ?.- I.6 Comments.- II The shift semigroup.- II.1 Introduction.- II.2 The prototype problem.- II.3 The dual space.- II.4 The adjoint shift semigroup.- II.5 The adjoint generator and the sun subspace.- II.6 The prototype system.- II.7 Comments.- III Linear RFDE as bounded perturbations.- III.1 The basic idea, followed by a digression on weak* integration.- III.2 Bounded perturbations in the sun-reflexive case.- III.3 Perturbations with finite dimensional range.- III.4 Back to RFDE.- III.5 Interpretation of the adjoint semigroup.- III.6 Equivalent description of the dynamics.- III.7 Complexification.- III.8 Remarks about the non-sun-reflexive case.- III.9 Comments.- IV Spectral theory.- IV.1 Introduction.- IV.2 Spectral decomposition for eventually compact semigroups.- IV.3 Delay equations.- IV.4 Characteristic matrices, equivalence and Jordan chains.- IV.5 The semigroup action on spectral subspaces for delay equations.- IV.6 Comments.- V Completeness or small solutions?.- V.l Introduction.- V.2 Exponential type calculus.- V.3 Completeness.- V.4 Small solutions.- V.5 Precise estimates for ??(z)-1?.- V.6 Series expansions.- V.7 Lower bounds and the Newton polygon.- V.8 Noncompleteness, series expansions and examples.- V.9 Arbitrary kernels of bounded variation.- V.10 Comments.- VI Inhomogeneous linear systems.- VI.1 Introduction.- VI.2 Decomposition in the variation-of-constants formula.- VI.3 Forcing with finite dimensional range.- VI.4 RFDE.- VI.5 Comments.- VII Semiflows for nonlinear systems.- VII.1 Introduction.- VII.2 Semiflows.- VII.3 Solutions to abstract integral equations.- VII.4 Smoothness.- VII.5 Linearization at a stationary point.- VII.6 Autonomous RFDE.- VII.7 Comments.- VIII Behaviour near a hyperbolic equilibrium.- VIII.1 Introduction.- VIII.2 Spectral decomposition.- VIII.3 Bounded solutions of the inhomogeneous linear equation.- VIII.4 The unstable manifold.- VIII.5 Invariant wedges and instability.- VIII.6 The stable manifold.- VIII.7 Comments.- IX The center manifold.- IX.1 Introduction.- IX.2 Spectral decomposition.- IX.3 Bounded solutions of the inhomogeneous linear equation.- IX.4 Modification of the nonlinearity.- IX.5 A Lipschitz center manifold.- IX.6 Contractions on embedded Banach spaces.- IX.7 The center manifold is of class Ck.- IX.8 Dynamics on and near the center manifold.- IX.9 Parameter dependence.- IX.10 A double eigenvalue at zero.- IX.11 Comments.- X Hopf bifurcation.- X.l Introduction.- X.2 The Hopf bifurcation theorem.- X.3 The direction of bifurcation.- X.4 Comments.- XI Characteristic equations.- XI.1 Introduction: an impressionistic sketch.- XI.2 The region of stability in a parameter plane.- XI.3 Strips.- XI.4 Case studies.- XI.5 Comments.- XII Time-dependent linear systems.- XII.1 Introduction.- XII.2 Evolutionary systems.- XII.3 Time-dependent linear RFDE.- XII.4 Invariance of X?: a counterexample and a sufficient condition.- XII.5 Perturbations with finite dimensional range.- XII.6 Comments.- XIII Floquet Theory.- XIII.1 Introduction.- XIII.2 Preliminaries on periodicity and a stability result.- XIII.3 Floquet multipliers.- XIII.4 Floquet representation on eigenspaces.- XIII.5 Comments.- XIV Periodic orbits.- XIV.1 Introduction.- XIV.2 The Floquet multipliers of a periodic orbit.- XIV.3 Poincaré maps.- XIV.4 Poincaré maps and Floquet multipliers.- XIV.5 Comments.- XV The prototype equation for delayed negative feedback: periodic solutions.- XV.1 Delayed feedback.- XV.2 Smoothness and oscillation of solutions.- XV.3 Slowly oscillatin

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